The present paper gives some results for the logarithm of the Riemann zeta-function and its iterated integrals. We obtain a certain explicit approximation formula for these functions. The formula has some applications, which are related with the value distribution of these functions and a relation between prime numbers and the distribution of zeros in short intervals.
We consider iterated integrals of {\log\zeta(s)} on certain vertical and horizontal lines.
Here, the function {\zeta(s)} is the Riemann zeta-function.
It is a well-known open problem whether or not the values of the Riemann zeta-function on the critical line
are dense in the complex plane.
In this paper,
we give a result for the denseness of the values of the iterated integrals on the horizontal lines.
By using this result,
we obtain the denseness of the values of {\int_{0}^{t}\log\zeta(\frac{1}{2}+it^{\prime})\,dt^{\prime}} under the Riemann Hypothesis.
Moreover, we show that, for any {m\geq 2}, the denseness of the values of an m-times iterated integral on the critical line is equivalent to the Riemann Hypothesis.
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